Research Article  Open Access
Chi Xu, Yu Jin, Duli Yu, "A Novel SigmaDelta Modulator with FractionalOrder Digital Loop Integrator", Mathematical Problems in Engineering, vol. 2017, Article ID 9861383, 7 pages, 2017. https://doi.org/10.1155/2017/9861383
A Novel SigmaDelta Modulator with FractionalOrder Digital Loop Integrator
Abstract
This paper proposes using a fractionalorder digital loop integrator to improve the robust stability of SigmaDelta modulator, thus extending the integerorder SigmaDelta modulator to a nonintegerorder (fractionalorder) one in the SigmaDelta ADC design field. The proposed fractionalorder SigmaDelta modulator has reasonable noise characteristics, dynamic range, and bandwidth; moreover the signaltonoise ratio (SNR) is improved remarkably. In particular, a 2ndorder digital loop integrator and a digital controller are combined to work as the fractionalorder digital loop integrator, which is realized using FPGA; this will reduce the ASIC analog circuit layout design and chip testing difficulties. The parameters of the proposed fractionalorder SigmaDelta modulator are tuned by using swarm intelligent algorithm, which offers opportunity to simplify the process of tuning parameters and further improve the noise performance. Simulation results are given and they demonstrate the efficiency of the proposed fractionalorder SigmaDelta modulator.
1. Introduction
SigmaDelta modulator technology has been commonly used in various fields including inertial sensors, such as the MEMS accelerometer and gyroscope [1–3]. SigmaDelta modulator not only converts analog signal to digital signal but also can suppress the quantization noise of bandwidth effectively. Most of SigmaDelta modulators are used in closedloop architect in terms of linearity, dynamic range, and bandwidth [4]. Considerable attention has been given to the issue of the MEMS accelerometer performance comparison between the loworder and highorder closedloop SigmaDelta modulators. Many literatures have pointed out that the loworder SigmaDelta modulator yields better stability with simpler design parameters, but the performance in terms of noise level is unsatisfactory. By contrast, the highorder SigmaDelta modulator has reasonable noise characteristics, dynamic range, and bandwidth, but its stability is not guaranteed. So, designing a SigmaDelta modulator with high stability and outstanding performance is much desired to be researched. Recently, highorder SigmaDelta modulator, which uses additional electronic integrator, has been designed [5]. However, the stability of the highorder SigmaDelta modulator is affected by variations of the MEMS accelerometer parameters [6]. Previous work has mainly focused on using integerorder integrator as a loop integrator to form the highorder SigmaDelta modulator to improve its SNR and noise performance [7]. In this paper, a novel SigmaDelta modulator with the fractionalorder digital loop integrator is discussed. The proposed novel SigmaDelta modulator contains a fractionalorder controller, which is able to provide the stability in the placement of fractionaltype poles and zeros.
Fractionalorder calculus belongs to the branch of mathematics, which is concerned with differentiations and integrations of nonintegerorder [8, 9]. According to [10], the remarkable advantage of fractionalorder integrator over its counterpart, the integerorder one, is that the stability and robustness of the fractionalorder integrator are much stronger. At present, modeling realworld phenomena using fractionalorder calculus has received great attention. As we all know, SigmaDelta modulators that have been applied so far were all considered as integerorder modulators, whereas the proposed SigmaDelta modulator with the fractionalorder digital loop integrator combines some characteristics of systems between the orders and (). Therefore, we will have more possibilities for an adjustment of the poles or zeros of the noiseshaping integrator according to special requirements through changing the system order as a real (not only integer) value. In this paper, a novel SigmaDelta modulator with the fractionalorder digital loop integrator is presented, where the fractionalorder digital loop integrator is cascaded between the analogfrontend amplifier and the quantizer. In detail, a 2ndorder digital integrator is used to perform noiseshaping of quantization noise from the comparator to improve SNR, and furthermore a digital controller is immediately in series with the 2ndorder digital integrator to provide the weak or strong fractionaltype poles or zeros to improve the robust stability for the proposed fractionalorder SigmaDelta modulator. The parameters of the proposed fractionalorder digital loop integrator are tuned by using particle swarm optimization (PSO) algorithm, which is easy to optimize the digital loop integrator parameters.
The rest of this paper is organized as follows: in Section 2, a generalized structure of SigmaDelta modulator is introduced; in Section 3, the proposed fractionalorder SigmaDelta modulator is discussed; then PSO algorithm for fractionalorder SigmaDelta modulator is presented in Section 4. Simulation results of the proposed SigmaDelta modulator are demonstrated and analyzed in Section 5. Finally, conclusions are given in Section 6.
2. Mathematical Model of the SigmaDelta Modulator System
Before discussing the proposed fractionalorder SigmaDelta modulator, a general system block diagram of SigmaDelta modulator is shown in Figure 1, which illustrates a typical architect of SigmaDelta modulator.
In Figure 1, is the transfer function of the sensing element, which is a 2ndorder electromechanical integrator; is the gain of analogfrontend (AFE) amplifier; is a 2ndorder digital integrator, which is employed to perform noiseshaping of quantization noise from the 1bit comparator to improve SNR. So, Figure 1 presents a MEMS accelerometerbased 4thorder SigmaDelta modulator, where the 2ndorder digital integrator is inserted between AFE and 1bit comparator; is the quantizer gain; is the quantization noise of 1bit quantizer, and is the equivalent linear model of 1bit DAC feedback. A quasilinear model of the 1bit quantizer is presented in Figure 1, where the quantizer output is equal to the sum of quantization noise and quantizer input with a quantization gain .
The linearized dynamical equation of the sensing element can be expressed in the Laplace domain as follows:where is the proof mass, is the damping coefficient, and is the spring constant. If = 20 mg, and ; the frequency domain characteristic curve of is shown in Figure 2.
In Figure 1, is the continuous frequency response of digital loop integrator , which can be expressed aswhere and are coefficients of the numerator and denominator.
3. FractionalOrder SigmaDelta Modulator System
3.1. Mathematical Model of the Proposed SigmaDelta Modulator
In this paper, the proposed fractionalorder SigmaDelta modulator is built by using the fractionalorder digital loop integrator instead of . The fractionalorder digital loop integrator of the proposed SigmaDelta modulator is shown in Figure 3.
is the 2ndorder digital loop integrator, which can be written as where , , , , , and are the coefficients of .
is the digital controller, which can be written aswhere , , and , , and are the integralorder, differentialorder, and proportional, integral, and differential coefficients of , respectively.
Here, the architect of the proposed fractionalorder SigmaDelta modulator is presented in Figure 4.
From the system model shown in Figure 4, the signal transfer function (STF) and the noise transfer function () of the proposed fractionalorder SigmaDelta modulator can be written as and are the continuous frequency responses of the digital loop integrators and , respectively. Taking (3) and (4) in (5) and (6), STF, can be rewritten aswhere
To achieve high SNR and stability of the overall system, , , , , , , , , and in (7) will be optimized by using PSO algorithm in the following section.
3.2. Stability Analysis for the Proposed SigmaDelta Modulator
In this paper, we use Caputo definition for fractional derivative, which is given aswhere is an integer satisfying and is Gamma function. Formula (9) can be transformed into the transfer function in the Laplace domain (assuming zero initial conditions) as follows:where and are constants, is the fractional commensurate order, and . Formula (10) can be rewritten as
Formula (11) is always represented as the integerorder system model when . As shown in [10], when matrix is deterministic without uncertainty, the stability condition for formula (11) is clearly expressed by
By observation from (12), the stability region of the fractionalorder () system is boarder than that of the integratororder one. In this paper, the fractional commensurate order is set as 0.5 to broader the stability region of the proposed SigmaDelta modulator. In order to obtain the fractional commensurate order , here we set ; therefore (7) can be rewritten as
By observation from (13), we can see that the order of the proposed SigmaDelta modulator is 4.5th order and it belongs to commensurate fractionalorder system. Therefore, the stability region of the proposed 4.5thorder SigmaDelta modulator becomes , , as shown in Figure 5.
It can be seen from Figure 5 that the stability region of the proposed 4.5thorder SigmaDelta modulator is wider than that of the traditional integerorder one, the stability of which is only in the lefthalf plane.
4. PSO Algorithm for FractionalOrder SigmaDelta Modulator
4.1. An Introduction to PSO Algorithm
Particle swarm optimization (PSO) algorithm belongs to the global search method [11]. It is inspired by social behavior of bird flocking and swarm theory. In PSO algorithm, the potential solutions are named as particles, and each particle is regarded as a point in a Ddimensional space that adjusts “flying” according to its own flying experience. The th particle is presented as . The best previous position of the th particle is recorded and represented as . The index of the best particle among all the particles in the population is represented by the symbol best. The velocity of the position change for particle is presented as . The particles are manipulated according to the following equation:where is the inertia weight and and are two positive constants.
4.2. Parameter Optimization
Firstly, the Simulink model of the proposed 4.5thorder SigmaDelta modulator is developed as shown in Figure 6.
In Figure 6, is the discretization of the fractionalorder controller. One of the discretization methods of the controller is to use a Tustin operator to approximate the fractionalorder operator [12]. The general form of Tustin operator is given as follows: is the sample period; is the fractional order. Here, we use Continued Fractional Expansion (CFE) to realize :
The mechanism and steps of this approximation method are presented in [12]. For instance, are approximated as follows:
The parameters of the proposed 4.5thorder SigmaDelta modulator are listed in Table 1.

4.3. Steps of Parameter Optimization
(1)Selection of initial value of PSO algorithm: here; , , , , , , , , and in (13) are optimized by PSO algorithm; hence, we consider as the position vector of PSO algorithm for the proposed 4.5thorder SigmaDelta modulator. The individual numbers are 600 corresponding to dimension 9 for the proposed 4.5thorder SigmaDelta modulator, and the iteration is set as 25: and (2)Objective function: a typical objective for SigmaDelta is high SNR, which can be maximized and is calculated based on the power spectral density of the output bit stream. Therefore, considering the high SNR as the objective of PSO algorithm is reasonable. For each individual simulation, the SNR is calculated by a function “calcSNR” available through the Delta Sigma Toolbox for Matlab(3)Initializing and in the optimal range and calculating the objective function(4)Applying (14) to update and and then to update best(5)Stopping of the iteration once the termination condition is satisfied. Otherwise, procedure goes back to step ()
5. Simulation
5.1. Simulation Results Discussion
The oversampling ratio (OSR) needs to be specified in the proposed 4.5thorder SigmaDelta modulator. Here, we choose sample frequency of 128 kHz, and . In each individual procedure, the SNR will be calculated when the Simulink model is running.
In our numerical experiment, The proposed 4.5thorder SigmaDelta modulator achieved about 92 dB to 118 dB of SNR by yielding the different values for . The optimal values of are selected as in Table 2.

The corresponding SNR of the proposed 4.5thorder SigmaDelta modulator is 117.647 dB, which is improved 10x compared to only 4thorder SigmaDelta modulator. The PSD plot of the 4.5thorder SigmaDelta modulator is also built with selected coefficients and shown in Figure 7. Compared to the 4thorder SigmaDelta modulator, the proposed 4.5thorder SigmaDelta modulator not only performs better SNR but also has wider noise floor (bandwidth: 115 Hz in the 4.5thorder SigmaDelta modulator and 103 Hz in the 4thorder SigmaDelta modulator) and sharper slope (amplitude gain: 75.1 dB/decade in the 4.5thorder SigmaDelta modulator and 53.6 dB/decade in the 4thorder SigmaDelta modulator).
5.2. Stability and Robustness Analysis
The fractionalorder digital loop integrator of the proposed 4.5thorder SigmaDelta modulator can be normalized as . Taking the linearized parameters , , and and the major transfer function and and into (5), a fractionalorder signal transfer function (STF) of the proposed 4.5thorder SigmaDelta modulator can be obtained. The root locus models of the proposed 4.5thorder SigmaDelta modulator and 4thorder SigmaDelta modulator are built and shown in Figure 8.
(a) 4.5thorder SigmaDelta modulator
(b) 4thorder SigmaDelta modulator
Figure 8 shows that the proposed 4.5thorder SigmaDelta modulator is stable. In Figure 8(b), the root locus of the 4thorder SigmaDelta modulator passes through the righthalf plane, whereas the poles and zeros of the 4.5thorder SigmaDelta modulator, as shown in Figure 8(a), are allocated along the negative axis of the plane and the farthest pole achieves −1.2 × 10^{7}. This leads to better stability of the designed 4.5thorder SigmaDelta modulator.
Similar results are also observed at root locus for parametric yield errors, such as different spring constant and damping coefficient , induced by manufacturing tolerances of MEMS. For instance, taking the spring constant = 1000 N/m in MEMS transfer function to verify the robust stability of the proposed 4.5thorder SigmaDelta modulator, Figure 9 shows the root locus of the proposed 4.5thorder SigmaDelta modulator with .
Also, taking = 2.4 × 10^{−2} Ns/m in MEMS transfer function, Figure 10 shows the root locus of the proposed 4.5thorder SigmaDelta modulator with = 2.4 × 10^{−2} Ns/m.
Simulation results show that the robust stability of the two sensing elements only presents an acceptable slight fluctuation, which indicates that the proposed 4.5thorder SigmaDelta modulator is robust to the sensitivity of MEMS devices.
6. Conclusions
A 4.5thorder SigmaDelta modulator structure is proposed in this paper. The proposed 4.5thorder SigmaDelta modulator achieves SNR = 117.647 dB in simulation and noise floor under −150 dB in frequency of 5–150 Hz. The simulated root locus shows improved stability compared to the pure integerorder system with good potential to progress further. This study can promote the development of high performance of MEMS accelerometer and also provides scientific and technical support for the application of fractionalorder theory to practical system.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the research fund to the top scientific and technological innovation team from Beijing University of Chemical Technology (no. buctylkjcx06).
References
 B. Lv, P. Wang, D. Wang, J. Liu, and X. Liu, “A highperformance closedloop fourthorder sigmadelta micromachined accelerometer,” Key Engineering Materials, vol. 503, pp. 134–138, 2012. View at: Publisher Site  Google Scholar
 X.C. Luo and J. Feng, “A monolithic MEMS gyroscope interface circuit in 0.35μm CMOS,” Tien Tzu Hsueh Pao/Acta Electronica Sinica, vol. 42, no. 9, pp. 1868–1872, 2014. View at: Publisher Site  Google Scholar
 S. Long, Y. Liu, K. He, X. Tang, and Q. Chen, “116 dB dynamic range CMOS readout circuit for MEMS capacitive accelerometer,” Journal of Semiconductors, vol. 35, no. 9, Article ID 095004, pp. 1–5, 2014. View at: Publisher Site  Google Scholar
 M. Keller, A. Buhmann, M. Ortmanns, and Y. Manoli, “A method for the discretetime simulation of continuoustime sigmadelta modulators,” in Proceedings of the IEEE International Symposium on Circuits and Systems, ISCAS '07, pp. 241–244, New Orleans, LA, USA, May 2007. View at: Google Scholar
 V. P. Petkov and B. E. Boser, “Highorder electromechanical modulation in micromachined inertial sensors,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 53, no. 5, pp. 1016–1022, 2006. View at: Publisher Site  Google Scholar
 Y. Dong, M. Kraft, and C. O. Gollasch, “A high performance accelerometer with fifth ordermodulator,” in Proceedings of the 15th MicroMechanics Europe Workshop, pp. 41–44, MEW, Leuven, BE, 2004. View at: Google Scholar
 C. Lang and R. Tielert, “A low noise accelerometer with digital pidtype controller and multibit force feedback,” in Proceedings of the the 25th European SolidState Circuits Conference, ESSCIRC '99, pp. 250–253, IEEE, Duisburg, Germany, 1999. View at: Google Scholar
 M. Caputo, “Distributed order differential equations modeling dielectric induction and diffusion,” Fractional Calculus and Applied Analysis, vol. 15, no. 4, pp. 421–442, 2001. View at: Google Scholar
 Y. Chen, H. Ahn, and D. Xue, “Robust controllability of interval fractional order linear time invariant systems,” in Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 1–9, ASME, Long Beach, Calif, USA, 2005. View at: Google Scholar
 M. Moze and J. Sabatier, “LMI tools for stability analysis of fractional systems,” in Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 1–9, ASME, Boston, MA, USA, 2005. View at: Google Scholar
 J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE International Conference on Neural Networks (ICNN '95), vol. 4, pp. 1942–1948, Perth, Western Australia, NovemberDecember 1995. View at: Publisher Site  Google Scholar
 C. Tricaud and Y. Chen, “An approximate method for numerically solving fractional order optimal control problems of general form,” Computers & Mathematics with Applications. An International Journal, vol. 59, no. 5, pp. 1644–1655, 2010. View at: Publisher Site  Google Scholar  MathSciNet
Copyright
Copyright © 2017 Chi Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.